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Decreasing minimization on base-polyhedra: relation between discrete and continuous cases

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Abstract

This paper is concerned with the relationship between the discrete and the continuous decreasing minimization problem on base-polyhedra. The continuous version (under the name of lexicographically optimal base of a polymatroid) was solved by Fujishige in 1980, with subsequent elaborations described in his book (1991). The discrete counterpart of the dec-min problem (concerning M-convex sets) was settled only recently by the present authors, with a strongly polynomial algorithm to compute not only a single decreasing minimal element but also the matroidal structure of all decreasing minimal elements and the dual object called the canonical partition. The objective of this paper is to offer a complete picture on the relationship between the continuous and discrete dec-min problems on base-polyhedra by establishing novel technical results and integrating known results. In particular, we derive proximity results, asserting the geometric closeness of the decreasingly minimal elements in the continuous and discrete cases, by revealing the relation between the principal partition and the canonical partition. We also describe decomposition-type algorithms for the discrete case following the approach of Fujishige and Groenevelt.

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Acknowledgements

We thank Satoru Iwata and Akiyoshi Shioura for discussion about algorithms, and Arie Tamir for indicating references. We also thank the anonymous referee for helpful comments. Insightful questions posed by Tamás Király led to Theorem 4.2. The research was partially supported by the National Research, Development and Innovation Fund of Hungary (FK_18) – No. NKFI-128673, and by JSPS KAKENHI Grant Number JP20K11697.

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Authors and Affiliations

  1. MTA-ELTE Egerváry Research Group, Department of Operations Research, Eötvös University, Pázmány P. s. 1/c, Budapest, 1117, Hungary

    András Frank

  2. The Institute of Statistical Mathematics, Tokyo, 190-8562, Japan

    Kazuo Murota

  3. Faculty of Economics and Business Administration, Tokyo Metropolitan University, Tokyo, 192-0397, Japan

    Kazuo Murota

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  1. András Frank
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  2. Kazuo Murota
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Correspondence to Kazuo Murota.

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Frank, A., Murota, K. Decreasing minimization on base-polyhedra: relation between discrete and continuous cases. Japan J. Indust. Appl. Math. 40, 183–221 (2023). https://doi.org/10.1007/s13160-022-00511-4

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  • DOI: https://doi.org/10.1007/s13160-022-00511-4

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